J. L. Bassani and V. Racherla Mechanical Engineering and Applied Mechanics

1. Mechanics of Materials Effects of Non-Glide Stresseson Plastic Flow and Failure Mechanismsarising from Non-Planar Dislocation Core Structures:Multiscale Simulations of Non-Associated Flow J. L. Bassani and V. Racherla Mechanical Engineering and Applied Mechanics V. Vitek and R. Groger Materials Science and Engineering University of Pennsylvania Support: NSF/ITR DMR-0219243 June 2004

2. Single Crystal Dislocation Dynamics & Crystal Plasticity Polycrystals Dislocation Core Homogenization & Finite Elements Atomistics - MD Multiscale Simulations of Non-Associated Plastic Flow Component Response • Atomistic studies of defect structures are the basis of models at progressively higher length-scales which ultimately are used to study macroscopic response and, in particular, failure mechanisms. • Our strategy is to pass only the most essential information on to higher length scales. Macroscopic Simulations identification of slip planes & non-glide stress components effective macroscopic behavior multi-slip models

3. OVERVIEW • Using accurate potentials to describe the atomic interactions of BCC metals and intermetallic compounds we have: • studied the influence of the stress state, on the motion of a screw dislocation from atomistic simulations • developed yield criteria for dislocation motion – effects of non-glide stresses on the Peierls barrier • developed multi-slip constitutive equations for single crystals • calculated yield surfaces and flow potentials for polycrystals • derived yield and flow functions for use in non-associated flow continuum models of polycrystals • studied implications of non-associated flow on bifurcation modes, forming limits, and cavitation instabilities

4. Core under pure shear on plane in [111] direction ( ) 1/2[111] Screw Dislocation Core in Mo: Transformations under Applied Stress Relaxed Structure: 3-fold symmetry Planes belonging to the [111] zone Core under pure shear perpendicular to Burgers vector – alone this stress cannot move the dislocation

5. (tM) x x x x x x Screw Dislocation under Shear Stress on MRSS Plane Molybdenum: for the dislocation moves on the plane at an a applied stress (on the MRSSP), but the atomistic results (circles) do not follow Schmid’s law: Other components of shear stress parallel to the Burgers vector affect the dislocation motion, and these can be expressed as a linear combination of the Schmid stress and one other shear stress, e.g., atomistic results

6. Motion of a Screw Dislocation in an Infinite Medium Using Bond Order Potentials

7. stress tensor Schmid stress on slip system a : (thermodynamic stress) slip plane normal slip direction non-glide stress components: where are crystallographic vectors that resolve each of the h =1,Nngnon-glide stress components that transform dislocation core structures (both shear and pressure). effective hardness yield criteria: (slip systema) non-glide stress yield parameters effective yield (Peierls) stress Yield Criteria with non-Glide Stresses identified from atomistic simulations

8. Slip Systems for BCC Crystals: the effects of non-glide shear stresses parallel and perpendicular to the Burgers vector yield criteria: Schmid stress: non-glide stresses:

9. Multiple Slip in Single Crystals with Non-Glide Stress Effects yield criteria: kinematics: a non-associated flow theory flow rule: (n >>1) BCC yield surface (a=0.6) – restricted model Two-dimensional projection of the yield surface. Euler angles for this crystal orientation are h, b, f = (0.785, 2.53, 0)

10. Single Crystal Dislocation Dynamics & Crystal Plasticity Polycrystals Dislocation Core Homogenization & Finite Elements Atomistics - MD Multiscale Simulations of Non-Associated Plastic Flow Component Response • Atomistic studies of defect structures are the basis of models at progressively higher length-scales which ultimately are used to study macroscopic response and, in particular, failure mechanisms. • Our strategy is to pass only the most essential information on to higher length scales. Macroscopic Simulations identification of slip planes & non-glide stress components effective macroscopic behavior multi-slip models

11. Random BCC Polycrystal with Non-Glide Stresses Consider a polycrystal of randomly oriented BCC grains each satisfying the yield criteria: . Neglecting elastic strains and assuming the strain in each crystal is the same as the macroscopic strain (Taylor hypothesis), a quadratic programming problem is used to solve for the minimum of 5 slips in each crystal, which gives an upper bound to the limit yield surface. For Schmid behavior (a=0) the classical Taylor factor is 3.07 times the slip-system yield stress in tension and compression.

12. 2D Yield Surfaces for Random BCC Polycrystals based upon single crystal yield criteria that include the effects of non-glide stresses: whereaandtoare material parameters

13. yield surface flow potential Non-Associated Flow Behavior macroscopic (engineering) theory F=G for classical associated flow behavior plastic strain-rate a = 0.6 These isotropic surfaces shown are predicted from a Taylor model of a random BCC polycrystal with single crystal yield criteria fitted to atomistic calculation of the stress-state dependence of the Peierls barrier in molybdenum.

14. Yield stress in compression Yield stress in tension note: reduces to the von Mises surface Macroscopic Yield Functions for Random BCCPolycrystals solid curve (best fit): points plotted are from Taylor calculation for BCC polycrystal with the effects of non-glide stresses both parallel to and perpendicular to the Burgers vector

15. Macroscopic Flow Potentials for Random BCC Polycrystals G is the flow potential: hardening law: solid curve (best fit): points plotted are from Taylor calculation for BCC polycrystal

16. e2 c e1 Effects of Non-Associated Flow on Bifurcations from Homogeneous Plane Stress Loading States If ∆D represents the jump in strain rate across the band (shown in red) and C the incremental modulus. The bifurcation condition is given by ∆D:C:∆D = 0

17. Effects of Non-Associated Flow on Forming Limits Sheet necking under biaxial straining MK analysis of sheet necking using deformation theory for strain hardening coefficients of N = 0.1.

18. Effects of Non-Associated Flow on Cavitation Instabilities critical pressure at unstable cavity growth p b

19. CONCLUSIONS • From the multiscale simulations beginning with the input from atomistics we observe that the non-glide stresses have similar order-of-magnitude effects at single and polycrystal levels and generally on macroscopic response. • Since these effects have their origin in dislocation core transformations, they arise generally at high stress levels, particularly at high strain-rates and/or low temperatures. • There are comparable order-of-magnitude effects on strain localization in the form of bifurcations, sheet necking, and on cavitation instabilities to name a few. • In the language of continuum plasticity, at each scale a significant effect of non-associated flow behavior is present.